Knowledge that Transforms

This paper derives the exact finite sample distribution of the two-stage generalized least squares (GLS) estimator in a multivariate linear model with general linear parameter restrictions. This includes the seemingly unrelated regression (SUR) model as a special case and generalizes presently known exact results for the latter system. The usual classical assumptions are made concerning nonrandom exogenous variables and normally distributed errors. The theoretical results of this paper are made possible by the author's development of a matrix fractional calculus. This operator calculus is the main theoretical tool of the paper and may be used to solve a wide range of other unsolved problems in econometric distribution theory. IN THE EARLY 1960's Zellner [10] developed a two-stage GLS estimator for the coefficients in a linear multivariate system that is now popularly known as the SUR model. This two-stage procedure has since been used in many empirical applications. GLS also forms the basis of other commonly used estimators both in linear models with heteroscedastic or autocorrelated errors and in simultaneous equation systems where it leads to three stage least squares (3SLS). In spite of extensive research and perhaps surprisingly in view of the popularity of GLS methods in empirical work, the exact finite sample distribution of the SUR estimator is known only in highly specialized cases. These cases effectively restrict attention to two equation systems and models with orthogonal regressors [2]. Existing distribution theory is even more limited in the case of other commonly used GLS estimators, such as the two-stage estimator in linear models with heteroscedastic errors. Here, only low order moment formulae are known and then only in the simplest two sample setting. The research underlying the present paper is motivated by the deficiencies outlined above. Our initial object of study was the exact distribution of the SUR estimator in the general case. But the methods we have developed open the way to an exact distribution theory for econometric estimators in a much wider setting than the SUR model. The present paper will derive the exact finite sample distribution of the two-stage GLS estimator in the multivariate linear model subject to general linear parameter restrictions. This generalizes all presently known distribution theory for the SUR model itself. Two important specializations of our results will be illustrated in detail: the unrestricted multivariate linear model; and the Zellner model with pairwise orthogonal regressors. The analytical results reported here are made possible by the introduction of a fractional matrix calculus. This calculus is developed in terms of the action of

[This paper considers a repeated principal agent relationship where the principal is risk neutral, the agent is risk averse, the principal can borrow or save at a fixed interest rate, and the agent discounts future consumption. It is shown that memory plays a very strong role in every Pareto-optimal contract. Sufficient conditions for Pareto-optimal contracts to exhibit rising or falling wages are identified. Finally, it is shown that the restriction of the agent's access to credit is necessary to achieve a Pareto-optimal outcome. In particular, under every Pareto-optimal contract for every outcome of every period the agent would choose to save some of his wage if he could.]

[In this paper we consider the computation of an equilibrium in a pure exchange economy with a block diagonal pattern. Such a structure appears in, e.g., international trade models. Utilizing the block diagonal pattern, the full equilibrium problem is reformulated as a problem on the product space of several lower-dimensional price-simplices. Some numerical results show the usefulness of this method.]

Expected value maximizing sequential search rules can be expressed in terms of reservation values. In search with learning the reservation value at any stage of the search is unknown until that stage is reached. Thus calculating ex ante (and subsequent) probabilities of search duration and the offer accepted is difficult if these probabilities are expressed in terms of reservation values. This paper shows, for a wide class of learning procedures, how re-expressing these probabilities in terms of fixed points allows their direct calculation and, thereby, calculation of the expected value of adaptive search. Examples and comparative statics results are presented.

[A dynamic model of insider trading with sequential auctions, structured to resemble a sequential equilibrium, is used to examine the informational content of prices, the liquidity characteristics of a speculative market, and the value of private information to an insider. The model has three kinds of traders: a single risk neutral insider, random noise traders, and competitive risk neutral market makers. The insider makes positive profits by exploiting his monopoly power optimally in a dynamic context, where noise trading provides camouflage which conceals his trading from market makers. As the time interval between auctions goes to zero, a limiting model of continuous trading is obtained. In this equilibrium, prices follow Brownian motion, the depth of the market is constant over time, and all private information is incorporated into prices by the end of trading.]

Before 1979, unemployment insurance (UI) benefits were not treated as taxable income in the United States. Several economists criticized this policy on the ground that not taxing UI benefits while taxing earned income allegedly encourages unemployed persons to conduct longer than socially optimal job searches. Since 1979, however, UI benefits received by persons in higher-income families have been subject to income tax. This paper investigates whether the introduction of benefit taxation has had the predicted effect of reducing unemployment duration.The study uses data on a sample of persons that filed for UI in 1978 or 1979 to examine whether high-income claimants collected benefits fo rshorter periods after the tax change than they did before benefits became taxable. As part of the empirical analysis, the paper develops a generalization of the Weibull distribution and applies a limited-dependent-variable technique for this distribution similar to the Tobit technique for the normal distribution. Despite some variation in the results from different model specifications, the analysis presents persuasive evidence of a tax effect on unemployment duration. The 1979 policy change is estimated to have reduced average compensated unemployment duration among the sampled high-income claimants by about one week.

[We consider an oligopolistic market where firms face an uncertain demand for their product. Each firm observes a private signal for the state of demand and decides whether to reveal it to other firms and how complete this revelation will be. After the stage of information transmission the firm chooses its level of output. We derive pure strategy equilibria that are symmetric and subgame perfect, and demonstrate that no information sharing is the unique Nash equilibrium of the game regardless of the degree of correlation among the private signals.]

Revolution is viewed as a two person game, between Lenin and the Tsar, who compete for support of coalitions of the population. The payoff is the probability of revolution, which Lenin seeks to maximize and the Tsar to minimize. Lenin's strategies are income distribution proposals; the Tsar's strategies are lists of penalties which members of the population will pay should they join Lenin and their bid for revolution fail. The probabilities of revolution depend on the strategies which the two revolutionary entrepreneurs propose. There is an equilibrium pair of strategies; the task is to study what properties it has. In particular, it is shown that various tyrannical aspects of the Tsar's strategy, and progressive aspects of Lenin's strategy need not flow from ideological precommitments, but are simply good optimizing behavior, given their respective goals in this game. Thus apparently ideological positions of Lenin and the Tsar are provided with microfoundations of a sort. The paper thus aims to: (i) study revolutions as strategic games, and more generally to (ii) be a case study of the rational evolution of apparently ideological behavior. 1. INTROI)UCTION REVOLUTIONS HAVE BEEN VIEWED by social scientists and historians, for the most part, as largely inexplicable events. According to the logic of collective (Olson [4]) the free rider problem should prevent each participant from joining in a revolutionary struggle. The side payments which might overcome such self-interested behavior are generally not offered in revolutionary situations. Rosa Luxemburg wrote of the psychology of the mass strike which made revolutionary events possible (Luxemburg [2]): contemporary students of rational action might characterize that psychology as the adoption, by the participants, of assurance game in contrast to prisoner dilemma game (Sen [7]). In the assurance game, an agent would rather cooperate (in this case revolt) if others do, rather than take a free ride. Economists, when they consider revolutions at all, view them as exogenous events, perhaps because they are so difficult to explain from rational self-interested behavior. Sociologists and political scientists have described which classes tend to be revolutionary (as opposed to reformist) in historical situations (for instance, Paige [6]), Skocpol [8], Stinchcombe [9]); these works show that any understanding of the formation of revolutionary preferences must be deeply rooted in the specifics of how the people involved earn their livelihood and how they interact. The present paper does not study these sociological aspects of the revolutionary situation; indeed, the sociology is taken as given and is embedded in various probability functions which are postulated. Nor do different production classes exist here, in the sense of groups of agents relating differently to the means of production. (I will, however, refer to different income classes in the paper.) Revolution is treated here as an allocation problem, a redistribution problem. The key actors upon whom attention will be focused are not the masses of people,

[For many multivariate linear rational expectations models a symmetry condition is satisfied, and when it is, a convenient representation of the solution exists that has many of the desirable features of the solution to the univariate model. This allows for greater insight into the identification, stability, and comparative dynamics properties of the model.]